The logic of distributive bilattices
Bilattices, introduced by Ginsberg (1988, Comput. Intell., 265–316) as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron (1996, J. Logic Lang. Inform., 5, 25–63) developed a logical system based...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2011 |
| País: | España |
| Institución: | Universidad Nacional de Educación a Distancia |
| Repositorio: | e-spacio. Repositorio Institucional de la UNED |
| Idioma: | inglés |
| OAI Identifier: | oai:e-spacio.uned.es:20.500.14468/19441 |
| Acceso en línea: | https://hdl.handle.net/20.500.14468/19441 |
| Access Level: | acceso abierto |
| Palabra clave: | Bilattice many-valued logic abstract algebraic logic reduced model non-protoalgebraic logic algebraizable Gentzen system. |
| id |
ES_4ea070daa655bea37ac53d66386d264f |
|---|---|
| oai_identifier_str |
oai:e-spacio.uned.es:20.500.14468/19441 |
| network_acronym_str |
ES |
| network_name_str |
España |
| repository_id_str |
|
| spelling |
The logic of distributive bilatticesBou, FélixRivieccio, UmbertoBilatticemany-valued logicabstract algebraic logicreduced modelnon-protoalgebraic logicalgebraizable Gentzen system.Bilattices, introduced by Ginsberg (1988, Comput. Intell., 265–316) as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron (1996, J. Logic Lang. Inform., 5, 25–63) developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of abstract algebraic logic (AAL). We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú (1995, Algebraizable Gentzen Systems and the Deduction of Theorem for Gentzen Systems) (even if the logic itself is not algebraizable). We also prove some purely algebraic results concerning bilattices, for instance that the variety of (unbounded) distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron (1996, Math. Struct. Comput. Sci., 6, 287–299) stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices (of which distributive bilattices are a proper subclass).Oxford University Presse-Spacio UNED20242024-05-2120112011-01-0120112011-01-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14468/19441reponame:e-spacio. Repositorio Institucional de la UNEDinstname:Universidad Nacional de Educación a DistanciaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/4.0/deed.esoai:e-spacio.uned.es:20.500.14468/194412026-06-06T12:38:31Z |
| dc.title.none.fl_str_mv |
The logic of distributive bilattices |
| title |
The logic of distributive bilattices |
| spellingShingle |
The logic of distributive bilattices Bou, Félix Bilattice many-valued logic abstract algebraic logic reduced model non-protoalgebraic logic algebraizable Gentzen system. |
| title_short |
The logic of distributive bilattices |
| title_full |
The logic of distributive bilattices |
| title_fullStr |
The logic of distributive bilattices |
| title_full_unstemmed |
The logic of distributive bilattices |
| title_sort |
The logic of distributive bilattices |
| dc.creator.none.fl_str_mv |
Bou, Félix Rivieccio, Umberto |
| author |
Bou, Félix |
| author_facet |
Bou, Félix Rivieccio, Umberto |
| author_role |
author |
| author2 |
Rivieccio, Umberto |
| author2_role |
author |
| dc.contributor.none.fl_str_mv |
e-Spacio UNED |
| dc.subject.none.fl_str_mv |
Bilattice many-valued logic abstract algebraic logic reduced model non-protoalgebraic logic algebraizable Gentzen system. |
| topic |
Bilattice many-valued logic abstract algebraic logic reduced model non-protoalgebraic logic algebraizable Gentzen system. |
| description |
Bilattices, introduced by Ginsberg (1988, Comput. Intell., 265–316) as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron (1996, J. Logic Lang. Inform., 5, 25–63) developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of abstract algebraic logic (AAL). We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú (1995, Algebraizable Gentzen Systems and the Deduction of Theorem for Gentzen Systems) (even if the logic itself is not algebraizable). We also prove some purely algebraic results concerning bilattices, for instance that the variety of (unbounded) distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron (1996, Math. Struct. Comput. Sci., 6, 287–299) stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices (of which distributive bilattices are a proper subclass). |
| publishDate |
2011 |
| dc.date.none.fl_str_mv |
2011 2011-01-01 2011 2011-01-01 2024 2024-05-21 |
| dc.type.none.fl_str_mv |
journal article http://purl.org/coar/resource_type/c_6501 |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/article |
| format |
article |
| dc.identifier.none.fl_str_mv |
https://hdl.handle.net/20.500.14468/19441 |
| url |
https://hdl.handle.net/20.500.14468/19441 |
| dc.language.none.fl_str_mv |
Inglés eng |
| language_invalid_str_mv |
Inglés |
| language |
eng |
| dc.rights.none.fl_str_mv |
open access http://purl.org/coar/access_right/c_abf2 info:eu-repo/semantics/openAccess https://creativecommons.org/licenses/by-nc-nd/4.0/deed.es |
| rights_invalid_str_mv |
open access http://purl.org/coar/access_right/c_abf2 https://creativecommons.org/licenses/by-nc-nd/4.0/deed.es |
| eu_rights_str_mv |
openAccess |
| dc.format.none.fl_str_mv |
application/pdf |
| dc.publisher.none.fl_str_mv |
Oxford University Press |
| publisher.none.fl_str_mv |
Oxford University Press |
| dc.source.none.fl_str_mv |
reponame:e-spacio. Repositorio Institucional de la UNED instname:Universidad Nacional de Educación a Distancia |
| instname_str |
Universidad Nacional de Educación a Distancia |
| reponame_str |
e-spacio. Repositorio Institucional de la UNED |
| collection |
e-spacio. Repositorio Institucional de la UNED |
| repository.name.fl_str_mv |
|
| repository.mail.fl_str_mv |
|
| _version_ |
1869407768536940544 |
| score |
15,812455 |